Gradient-type nonlinear resonance elliptic systems on the Sierpiński gasket

Abstract

This paper investigates resonant gradient-type elliptic systems defined on the Sierpiński gasket. We prove the existence of infinitely many nontrivial weak solutions involving the weak Laplacian operator with zero Dirichlet boundary conditions. The analysis is based on variational methods and critical point theory, suitably adapted to the fractal framework. We exploit the analytic and geometric properties of the gasket, including compact embeddings of the associated energy space. The nonlinear term satisfies resonance conditions linked to the spectrum of the Laplacian. Due to the non-smooth nature of the fractal domain, classical tools require refinement to ensure compactness and coercivity. Our results extend existing existence theorems for gradient systems to fractal domains, revealing new interactions between nonlinear analysis, variational methods, and geometric measure theory. This work enhances the understanding of elliptic problems on self-similar structures and contributes to the broader theory of PDEs in non-Euclidean settings.